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5.0 out of 5 stars
Another way to enjoy fundamental physics!, October 14, 2007
Arieh Ben-Naim, professor at the Hebrew University of Jerusalem, taught
thermodynamics and statistical mechanics for many years and is well
aware that students learn the second law but do not understand it,
simply because it can not be explained in the framework of classical
thermodynamics, in which it was first formulated by Lord Kelvin (i.e.
William Thomson, 1824-1907) and Rudolf Julius Emanuel Clausius
(1822-1888). Hence, this law and the connected concept of entropy are
usually surrounded by some mysterious halo: there is something (the
entropy), defined as the ratio between heat and temperature, that is
always increasing. The students not only do not understand _why_ it is
always increasing (it is left as a principle in classical
thermodynamics), but also ask themselves what is the _source_ of such
ever increasing quantity.
We feel comfortable with the first law, that is the principle of energy conservation, because our experience always
suggests that if we use some resource (the energy) to perform any work,
then we are left with less available energy for further tasks. The
first law simply tells us that the heat is
another form of energy so that nothing is actually lost, something which
we can accept without pain. In addition, the second law says that,
though the total energy is constant, we can not always recycle 100% of
it because there is a limit on the efficiency of conversion of heat into
work (the highest efficiency being given by the Carnot cycle, named
after Nicolas Léonard Sadi Carnot, 1796-1832). Again, we can accept it
quite easily, because it sounds natural, i.e. in accordance with our
common sense: we do not know any perpetual engine. But our daily
experience is not sufficient to make us understand what entropy is, and
why it must always increase.
The author shows that, if we identify the entropy with the concept of
"missing information" of the system at equilibrium, following the work
done by Claude Elwood Shannon (1916-2001) in 1948, we obtain a well
defined and (at least in principle) measurable quantity. This quantity,
apart from a multiplicative constant, has the same behavior as the
entropy: for every spontaneous process of an isolated system, it must
increase until the equilibrium state is reached. The missing
information, rather than the disorder (not being a well defined
quantity), is the key concept to understand the second law.
I should say here that the identity of entropy and missing
information is not a widespread idea among physicists, so that many
people may not appreciate this point. However, the arguments of this
book are quite convincing, and different opinions are also taken into
account and commented.
In addition, Ben-Naim thinks that the entropy should be taught as an
dimensionless quantity, being defined as the ratio between heat, that is
energy, and temperature, that is a measure of the average kinetic energy
of the atoms and molecules. The only difference with the missing
information, again dimensionless, is the scale: because the missing
information can be defined as the number of binary questions (with
answer "yes" on "no" only) which are necessary to identify the
microscopic state of the system, this number comes out to be incredibly
large for ordinary physical systems, involving a number of constituents
of the order of the Avogadro's number. This numerical difference makes
me think about the difference between mass and energy, connected by the
Einstein's most famous equation E = m c^2: they could be measured using
the same units (as it is actually done in high-energy physics), the sole
difference being that even a very small mass amounts to a huge quantity
of energy.
The mystery of the ever increasing entropy can be explained once (and
only if) we realize that the matter is not continue, but discrete. The
author basically follows the work of Josiah Willard Gibbs (1839-1903),
who developed the statistical mechanical theory of matter based on a
purely probabilistic approach. First, one has to accept the fact that
macroscopic measurements are not sensitive enough to distinguish
microscopic configurations when they differ for thousands or even
millions of atoms, just because the total number of particles is usually
very large (usually of the order of 10^23 at least). Then, under the
hypothesis that each microscopic state is equally probable, i.e. that
the system will spend almost the same time in each micro-state, one can
group indistinguishable micro-states into macro-states. The latter are
the only thing we can monitor with macroscopic measurements. Under the
commonly accepted hypothesis that all microscopic configurations are
equally probable, macro-states composed by larger numbers of
micro-states will be more probable, i.e. the system will spend more time
in such macro-states.
As a naive example, one could start with a system prepared in such a way
that all its constituents are in the same microscopic configuration.
One could think about a sample of N dices, all of them showing the same
face, say the first one. The questions could be: (1) "Are all dices
showing the same face?"--Yes--; (2) "Is the face value larger or equal
than 3?"--No--; (3) "Is the face value larger or equal than 2?"--No--;
at this point we know that the value is 1. In general, the number of
binary questions is proportional to the logarithm in base 2 of the
number C of possible configurations, that is O(log_2 C). Now imagine to
randomly mix the dice by throwing all of them. The answer to the first
question would be "No", so that a completely different series of
questions has to be asked to find the microscopic configurations.
First, one may procede by finding how many dice show the value 1, for
example, asking O(log_2 N) questions. Suppose that the answer is M<N:
then one should find exactly what dice are showing this face, by asking
O(N) questions. The next step is to find how many dice show the value
2, among the N-M remaining ones, and so on. When N is very large, the
number of questions increases rapidly. So far, we have being speaking
about "microscopic" configurations, describing the exact state of all
dice. Now, we can imagine to be interested only in the "macroscopic"
configuration defined by the sum of all values. It is very easy to
imagine that the "microscopic" configurations corresponding to sum
values around 3N (corresponging to a uniform distribution of values)
will be many more than those with sum near N or 6N (which need all dice
showing 1 or 6, respectively). If we repeatedly shake the box or throw
all dice, most of the time we will obtain a sum near to 3N, and larger
deviations will be rarer. Hence, such a system will soon approach the
"equilibrium" state in which the sum is very near to 3N.
As a matter of fact, when the number of possible microscopic
configurations increases, the probability distribution of macro-states
becomes narrower and narrower, so that for ordinary systems the
probability to have a fluctuation large enough to be measured is
incredibly small. Actually, as Ben-Naim clearly emphasizes, the
probabilistic formulation of the second law of thermodynamics allows us
to quantify its validity, in terms of the time one should wait to be
able to find a fluctuation large enough to be measured. It comes out
that, for ordinary systems, the probability to have any measurable
fluctuation away from the equilibrium state is so low that the universe
age is practically negligible compared to the time we should wait to
observe such fluctuation. From this point of view, the second law is
far more "absolute" than the other laws of physics, for which at best we
could state that they are valid since the beginning of the universe life.
The book is a very good reading for all students who approach the
thermodynamics and also for more advanced people who do or do not feel
comfortable with the fascinating concept of entropy. Ben-Naim is also
the author of a more technical book ("Statistical Thermodynamics Based
on Information. A Farewell to Entropy", World Scientific,
A Farewell To Entropy) in
which these guidelines are the base for a more detailed treatment of
statistical mechanics. Because we usually learn things much better when
following a cyclical approach, I encourage the readers to start with the
book "Entropy Demystified" and then seriously consider to go deeper into
the details of statistical mechanics with the more technical book by
Ben-Naim, of which I was delighted to read the draft.
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